Question 5
The vector equation of the line that passes through the point $A(1,\,2,\,-1)$ and is parallel to the vector $2\hat{\boldsymbol{\imath}}-\hat{\boldsymbol{\jmath}}+3\hat{\boldsymbol{k}}$, where $t$ is a scalar, is
That’s it. Step through the hints below to see why.
Not quite. You swapped the position vector and the direction vector.
Not quite. You made a sign error in the $\hat{\boldsymbol{k}}$ component of the position vector ($+\hat{\boldsymbol{k}}$ instead of $-\hat{\boldsymbol{k}}$).
Not quite. You read “parallel to” as “passes through” and used $\boldsymbol{d}-\boldsymbol{a}=\hat{\boldsymbol{\imath}}-3\hat{\boldsymbol{\jmath}}+4\hat{\boldsymbol{k}}$ as the direction.
Use $\boldsymbol{r}=\boldsymbol{a}+t\boldsymbol{d}$ with $\boldsymbol{a}=\overrightarrow{OA}$ and $\boldsymbol{d}$ the parallel vector.
$$\begin{aligned}\boldsymbol{r}=&\left(\hat{\boldsymbol{\imath}}+2\hat{\boldsymbol{\jmath}}-\hat{\boldsymbol{k}}\right)\\&+t\left(2\hat{\boldsymbol{\imath}}-\hat{\boldsymbol{\jmath}}+3\hat{\boldsymbol{k}}\right)\end{aligned}$$
$$\boldsymbol{r}=\left(\hat{\boldsymbol{\imath}}+2\hat{\boldsymbol{\jmath}}-\hat{\boldsymbol{k}}\right)+t\left(2\hat{\boldsymbol{\imath}}-\hat{\boldsymbol{\jmath}}+3\hat{\boldsymbol{k}}\right)$$
This is option (D).
Only mark this done when you could do it without help. Reading the solution does not count.
A Tangent Tuition mock exam question. It is not a past QCAA question, and Tangent Tuition is not affiliated with the QCAA.
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